Probabilistic Fitting of Topological Structure to Data

TL;DR

This paper introduces simplicial mixture models inspired by algebraic topology to fit topological structures to data, demonstrating their approximation capabilities and practical applications in data unmixing.

Contribution

It proposes a novel class of probability distributions that encode topological information and shows how to fit these models to data using maximum likelihood.

Findings

Models can approximate complex topologies with enough simplices.

Fitted models provide meaningful archetype vectors for data unmixing.

The approach is feasible and effective for capturing data topology.

Abstract

We define a class of probability distributions that we call simplicial mixture models, inspired by simplicial complexes from algebraic topology. The parameters of these distributions represent their topology and we show that it is possible and feasible to fit topological structure to data using a maximum-likelihood approach. We prove under reasonable assumptions that with a fixed number of vertices a distribution can be approximated arbitrarily closely by a simplicial mixture model when using enough simplices. Even if the topology is not of primary interest, when using a model that takes the topology of the data into account the vertex positions are good candidates for archetype/endmember vectors in unmixing problems.